The cardiovascular system is thought to be organized to efficiently transport oxygen, nutrients and hormones through the body. Yet, although micro-vessels, including capillaries and arterioles, account for half of the total dissipation within the network, they show few of the transport efficiency features that are observed in large vessels. We will develop a linked picture of the physics of healthy microvascular network function, from the geometry and connectivity of vessels to their dilation and response to occlusion, with the general goal of determining the physical principles that underlie the organization of healthy microvascular networks, as well as their capacity to reorganize following damage. To accomplish this goal we will develop a new mathematical toolkit of modeling, optimization and analysis, as well as directly testing the predictions of these models using the embryonic zebrafish model system. We aim to answer several fundamental questions about microvascular organization. First, what optimization principles and constraint functions shape the zebrafish microvasculature? Second, how do changes in conductivity affect global fluxes at different levels within the vascular network? Third, what role does dynamic red blood cell occlusion in maintaining uniform perfusion of surrounding tissues? Fourth, how nearly are target functions restored, following damage in disease and trauma models? In other words, which target functions are robust to damage, and which do not recover? New mathematical tools will be developed: specifically, the project will require new methods of optimization on spatially embedded networks, a problem that is associated with complex functionals with non-convex landscapes, many degrees of freedom, and in which the number of constraints scales with the size of the network. Resolving these problems will require rigorous analysis of the convergence of the optimization algorithm, of the effect of the mesh used to discretize the vascularized tissue upon this convergence, and development of multiscale methods to handle feedbacks from the finest scales, including occlusive feedbacks from individual red blood cells and the dilation of individual vessels, upon global partitioning of blood flows. These new tools will translate immediately into new fundamental biological insight into how microvascular networks are physically organized, and into how this physical organization is disrupted, and the extent to which it can be restored, following damage. The project therefore aligns with the objectives of the BBCB by creating broadly applicable mathematical and computational tools for quantifying and understanding microvascular physiology and health.